They argue: “A diagnosis of dementia is a life-changing event. The Government’s NHS Mandate states that: ‘Dementia is the illness most feared by people in England over the age of 55.’

“There is a very real danger that older patients will avoid visiting their doctor with significant health problems, for fear of being given a diagnosis they do not wish to have.”

They say the Dementia Case Finding Programme, as it is officially called, will lead to “many people being labelled as having ‘Mild Cognitive Impairment’ ".

This condition, characterised by some memory problems and occasional confusion, is often but not always a precursor to dementia.

This diagnosis could “cause them profound anxiety, ruining their quality of life”, say the doctors.

Drugs can help but are “no panacea”, and they worry that the diagnosis will cause more harm than good.

They continue: “The danger of this policy is that it will divert the resources needed to care properly for people with dementia, to people who are managing well and have minor cognitive changes that do not affect their lives.

“This is especially dangerous as people with severe dementia are least able to raise concerns about their care.”

The letter’s authors also maintain the programme runs counter to advice from the UK National Screening Committee.

It concluded in June 2010 that screening for Alzheimer’s - the most common form of dementia - “should not be offered”.

Dr Martin Brunet, a Surrey GP, who organised the letter, said they did not want to put people off consulting doctors if they thought they were exhibiting early signs of dementia.

But he said the Government’s proposal was different because it involved “actively seeking people [with early signs of dementia] who aren’t looking for help”.

He continued: “Our memory clinics, where elderly patients go for dementia assessments, are overstretched at the moment.

“The danger is, of swamping a stressed system with people who are not necessarily going to be helped by a diagnosis, and could be harmed.”

Dr Iona Heath, immediate former president of the Royal College of GPs, who signed the letter, said: "This is just another step on the road to obsessive diagnosis, and turning people into patients without any notion that it might cause more harm than good and divert resources."

She added: "When I was a medical student in the 1970s, we were taught that there was 'a lot of illness without disease, and not much disease without illness'. Now we are taught that there's an epidemic of unrecognised disease, but we don't see the symptoms."

Gill Phillips, a company director from Coventry, said she had concerns her elderly mother would be put off visiting her GP by what she described as a “covert” system of screening for dementia.

She said: “My Mum is 90 and very good mentally but she is terrified of dementia, as several of her friends are affected.

“If she knew that a visit to the GP would result in her being screened for dementia, I think it would stop her going to the GP at all.

“She has the idea that they would make her count backwards in sevens which she has always found difficult.

“She tells me to make sure that if she is ever tested, they give her The Daily Telegraph crossword instead.

“This has been a long-standing joke between us but suddenly seems serious. If she were ever to be screened for dementia, I would want to be there to ask questions and support her afterwards.

“I would not have this option under the proposed “covert” system.”

However, a group of doctors led by Dr Jill Rasmussen, a GP and advisor to the Alzheimer’s Society, said those objecting risked "fuelling the fire of stigma".

They said there were around 400,000 people with dementia “who may be being denied the opportunity to have a diagnosis with the support and potential treatment that can follow”.

In a response in BMJonline, they wrote: “We know that a diagnosis reduces symptoms of anxiety and depression in patients and their carers. To suggest it causes solely harm risks fuelling the fire of stigma.”

Andrew Chigley, of the Alzheimer’s Society, added: “It’s crucial that GPs are geared up to diagnose the condition as early as possible in individuals showing signs and symptoms.

“A proactive approach to assessing people at risk could be a crucial step forward to driving up diagnosis rates and improving lives.

“However, asking people at a high risk of dementia whether they’re having memory problems is not the same as population screening – something Alzheimer’s Society does not currently support.

“These proposals are now under consultation and its vital doctors and others share their views.”

A Department of Health spokesman said: "We want the UK to to be one of the best places for dementia care in Europe and are working towards making that a reality.

"For far too long, people with dementia and their carers have not received the care and support they deserve.

"Around 400,000 people are currently undiagnosed denying them support and treatment - which has to change.

"The proposals do not introduce screening for Alzheimer's disease, which we agree would be inappropriate, but are designed to pick up symptoms in those known to be at risk of dementia, such as people with Coronary Heart Disease, Stroke or Parkinson’s."